A Weak Structural Form of Commutative Equivalence in Finite Codes

Dean Kraizberg

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arXiv · 2603.27656

A Weak Structural Form of Commutative Equivalence in Finite Codes

Dean Kraizberg

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Abstract

We investigate the structural relationship between prefix-free codes over the binary alphabet and a class of unlabeled rooted trees, which we call \emph{symmetric} trees. We establish a canonical correspondence between prefix-free codes and symmetric trees, preserving not only the lengths of codewords but also some additional commutative structure. Using this correspondence, we provide a result related to the commutative equivalence conjecture. We show that for every code, there exists a prefix-free code such that, for each fixed word length, the sums of powers of two determined by the occurrences of a distinguished symbol are equal.

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Dean Kraizberg. 2026-03-29. A Weak Structural Form of Commutative Equivalence in Finite Codes. https://arxiv.org/abs/2603.27656

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